Putcha semigroups

Abstract

In [80], M.S. Putcha characterized semigroups which are decomposable into semilattice of archimedean semigroups. He showed that a semigroup S is a semilattice of archimeden semigroups if and only if, for every a, b ∈ S, the assumption a ∈ S ¹ bS ¹ implies a ⁿ ∈ S ¹ a ² S ¹ for some positive integer n. Semigroups with this condition are called Putcha semigroups. In this chapter we also consider the left Putca semigroups and the right Putcha semigroups (Definition 2.1). It is proved that a semigroup is a simple left and right Putcha semigroup if and only if it is completely simple. By the help of this result, the retract extension of completely simple semigroups by nil semigroups are characterized. It is shown that a semigroup is a retract extension of a completely simple semigroup by a nil semigroup if and only if it is an archimedean left and right Putcha semigroup containing at least one idempotent element.